In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 17 Jun., 19:36, MoeBlee <jazzm...@hotmail.com> wrote: > > Re: Jun 17, 12:23 pm, MoeBlee <jazzm...@hotmail.com>e: > > > > P.S. > > > > If the discussion about the formal theory ZFC, then I don't know what > > lack of trichotomy you're referring to. > > > > Of course, many relations fail to satisfy trichotomy. But the standard > > strictly less than ordering on the reals satisfies trichotomy: > > > > If x and y are reals, then exactly one of these holds: > > > > x <_r y > > y <_r x > > x=y > > > > where '<_r' stands for the standard stricly less than relation on the > > set of real numbers. > > That means, even undefinable real numbers are in trichotomy with each > other and in particular with definable real numbers?
But it does not mean that we can ALWAYS determine which of two reals is the larger.
> > OK. Let us stay with the reals. Say you have two undefinable reals > between 0 an 1. How can you manage to find out which one is less than > the other?
Perhaps you don't. That one of a limited number of options is the correct one does not necessarily mean that which one is correct can be determined.
In every sufficiently complex system, including those sufficiently complex to support standard arithmetic, there are statements whose truth cannot be determined within that system.